Parallel magnetic resonance imaging (pMRI) techniques reduce scan time by undersampling k-space; this directly improves temporal resolution in real-time cine imaging and other applications. GeneRalized Autocalibrating Partially Parallel Acquisition (GRAPPA), a k-space based pMRI technique, is widely used clinically for magnetic resonance imaging (MRI) because of its robustness. It estimates the missing k-space points by solving a set of linear equations; however, this method does not take advantage of the correlations within the missing k-space data. In reality, all k-space data within a neighborhood are correlated. These correlations can be formulated as additional self-constraint conditions, which are not considered in standard GRAPPA. While it has been established that incorporating these self-constraints in parallel reconstruction greatly improves the image quality, this has only been previously demonstrated using iterative solutions.
Conventional solutions utilize linear constraints with iterative solvers to improve the performance of GRAPPA reconstruction. For example, Zhao and Hu proposed to estimate a self-constraint kernel that utilizes the reconstructed lines to interpolate the acquired lines (iGRAPPA) (see, T. Zhao and X. Hu, “Iterative GRAPPA (iGRAPPA) for improved parallel imaging reconstruction,” Magn Reson Med 59, 903-907 (2008)). Lustig and Pauly proposed the SPIRiT technique, which exploits the correlations in k-space by applying a full neighborhood kernel (see, M. Lustig and J. M. Pauly, “SPIRiT: Iterative self-consistent parallel imaging reconstruction from arbitrary k-space,” Magn Reson Med 64, 457-471 (2010)). Zhang et al. proposed the PRUNO technique, which uses the null-space method to take advantage of k-space correlations (see, J. Zhang, C. Liu and M. E. Moseley, “Parallel reconstruction using null operations,” Magn Reson Med 66, 1241-1253 (2011)).
However, these methods do not have closed-form solutions, and the corresponding reconstruction problem can only be solved using iterative solvers. There are some ubiquitous difficulties associated with iterative methods. First, defining an appropriate stopping criterion can be problematic; second, convergence may not be guaranteed; and third, these methods are computationally demanding and real-time reconstruction may be infeasible.